SS3657 Lecture 5
I
Review of the basic properties of cdf functions.
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The properties of pdf and cdf of a continuous r.v.
I
Textbook coverage: Chap 2.2
1/9
Identify which of the following plots displays a valid cdf
(only ONE correct)
0.8
0.6
F(x)
0.4
0.2
0
SS3657 Lecture 1
Review old concepts with new perspective:
I Examples
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I
The throwing two dices for a sum of 7;
The birthday problem
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The concept of probability triplet (, F, P)
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The axioms of probability
I
Textbook coverage: Chapter 1: 1.2-1.4
1/8
Pro
SS3657 Lecture 2
It is all about conditional probability
I Examples
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I
The Polyas Urn model
The Monty hall problem
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The definition of conditional probability
I
The Law of Total Probability
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The Bayes Rule
I
Textbook coverage: Chapter 1: 1.5
1/9
The def
SS3657 Lecture 3
I
The concept of independence
I
I
I
I
Examples
I
I
I
Pairwise independence v.s. (mutually) independence
Physical independence v.s. statistical independence
Mutually exclusive events are actually dependent
The drawing a ball from 4 balls
T
SS3657 Lecture 4
I
The definition of random variables
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The properties of cdf F ()
Discrete r.v.s
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I
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cdf and pmf of a discrete r.v.
Textbook coverage: Chap 2.1
1/17
The definition of random variable (rv)
I
I
Motivation: why do we need random variable i
SS3657 Lecture 1
Review old concepts with new perspective:
I Examples in the form of clicker questions
I
I
The probability of obtaining a sum of 7 when throwing two
dices
The birthday problem
I
The concept of probability triplet (, F, P)
I
The axioms of p
SS3657 Lecture 3
I
I
Examples in the form of clicker questions: circuit example.
The concept of independence
I
I
I
I
Pairwise independence v.s. (mutually) independence
Physical independence v.s. statistical independence
Mutually exclusive events are actua
SS3657 Lecture 6
I
Transformations or functions of a r.v.
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Textbook coverage: Chap 2.3
1/14
Transformation Y = g(X)
I
Consider a transformation Y = g(X), where g is a function
g : D 7 R.
The domain D of g is typically the reals in R, but it must at
least
Urn Game Binary Tree Plots
Choose balls from an urn. The urn has R = 4 red balls and W = 6 white balls. Figure 1 shows the probability rules for a game without replacement and Figure 2 shows the probability rules for a game with replacement. Consider the
Change of Variables and Marginals: Example of Student's t Ratio
Suppose that X and Y are independent r.v.'s, X R and Y > 0. A specific example below is X N (0, 1) and Y 2 . (n) Consider the transformation of the r.v.'s V W where a > 0 is a positive number
Ratio of Two Independent Unif(-1,1) Random Variables
1
General Comments
Y Suppose that X, Y are iid Uniform(-1,1) random variables. Let Z = X . In this handout we find the pdf of Z using two methods. One is the method of the special form derived for the r
Order Statistics and Distributions
1
Some Preliminary Comments and Ideas
In this section we consider a random sample X1 , X2 , . . . , Xn common continuous distribution function F and probability density f . Thus Xi , i = 1, 2, . . . , n are i.i.d. F (or
University of Western Ontario Statistics 3657 Term Test I
October 19, 2009, 5:30 PM - 6:30 PM
Instructor: R. J. Kulperger
1 2 3 4 Total (50)
Name : ID :
1
Statistics 3657a Term Test, October 19, 2009, 5:30 PM - 6:20 PM
2
Instructions: I. Make sure that yo
University of Western Ontario Statistics 357 Term Test I
October 15, 2007, 5:30 PM - 6:30 PM
Instructor: R. J. Kulperger Instructions: I. Make sure that your name and ID number are the front page, and every page, of your question booklet. II. This term te
University of Western Ontario Statistics 3657 Term Test II
22 November, 2010
Instructor: R. J. Kulperger Instructions: I. Make sure that your name and ID number are the front page, and every page, of your question booklet. II. This term test is of 60 minu
University of Western Ontario Statistics 357 Term Test II
12 November, 2007
Instructor: R. J. Kulperger Instructions: I. Make sure that your name and ID number are the front page, and every page, of your question booklet. II. This term test is of 60 minut
Sample Mean and Variance for I.I.D. Normals
Suppose that Xi , i = 1, . . . , n are i.i.d. N (0, 1) random variables, n 2. The sample mean and variance are then given by X s2 = = 1 Xi n i=1
n
) 1 ( 2 Xi - X n - 1 i=1
n
Theorem 1 Under the condition above,
Statistics 3657 : Moment Approximations
1
Preliminaries
Suppose that we have a r.v. X and that we wish to calculate the expectation of g(X) for some function g. Of course we could calculate it as E(g(X) by the appropriate integral (continuous r.v. ) or su
Statistics 3657 : Moment Generating Functions
A useful tool for studying sums of independent random variables is generating functions. In this course we consider moment generating functions. Definition 1 (Moment Generating Function) Consider a distributio
Law of Large Numbers and Central Limit Theorem
1
Convergence in Probability and Law of Large Numbers
Definition 1 A sequence of r.v.s Xn , n = 1, 2, 3, . . . is said to converge to Y if and only if for every > 0 then P (|Xn - Y | > ) 0 . In this course we
SOME HELPFUL FORMULAE 1. Gamma function and properties: (a) () =
0
x-1 e-x dx
(b) For > 0, ( + 1) = (). (c) For integers n 1, (n) = (n - 1)!. (d) (1/2) = 2. Binomial, parameter and n the number of trials ( ) n p(k) = (1 - )n-k k , k = 0, 1, 2, . . . , n .
Moments or Expectation of Functions of Random Variables
Definition 1 For a discrete random variable X define the expected value of X as xP (X = x) E(X) =
x
provided this sum is well defined. The sum is well defined provided it is either finite, + or -. Th
University of Western Ontario Statistics 357 Final Exam
14 December, 2007: 7 PM - PM
Instructor: R. J. Kulperger Instructions: I. Make sure that your name and ID number are the front of your exam question sheet. II. All answers are to be written in this e
Statistics 3657 : Convergence
Definition 1 Let Xn be a sequence of r.v.s. We say Xn converges in probability to a constant a if and only if for any > 0 P (|Xn - a| > ) 0 Theorem 1 Suppose Xi is a sequence of iid random variables with mean and variance 2 .
Statistics 3657 : Conditional Expectation
See file ch3-3-eg1.tex for some conditional distributions and conditional expectations for a specific bivariate pdf. The conditional expectation of Y given X is defined to be a function of X, that is
E(Y |X) = h(X
Chapter 3.6 Transformation Examples
1
General Comments on Transformations and Distributions
This section studies how to find the distribution of Y = g(X), where Y may be of dimension 1 or higher and X may be of dimension 1 or higher. The function g:DE has
Joint Distribution Example
Toss 4 fair coins, and consider the two r.v.'s: X = number of H's (successes) in the first 3 coin tosses Y = number of H's (successes) in the last 2 coin tosses It is simplest for us to obtain the joint distribution of these two